现代傅里叶分析 第2版 英文PDF|Epub|txt|kindle电子书版本网盘下载

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6 Smoothness and Function Spaces1

6.1 Riesz and Bessel Potentials,Fractional Integrals1

6.1.1 Riesz Potentials2

6.1.2 Bessel Potentials6

Exercises9

6.2 Sobolev Spaces12

6.2.1 Definition and Basic Properties of General Sobolev Spaces13

6.2.2 Littlewood-Paley Characterization of Inhomogeneous Sobolev Spaces16

6.2.3 Littlewood-Paley Characterization of Homogeneous Sobolev Spaces20

Exercises22

6.3 Lipschitz Spaces24

6.3.1 Introduction to Lipschitz Spaces25

6.3.2 Littlewood-Paley Characterization of Homogeneous Lipschitz Spaces27

6.3.3 Littlewood-Paley Characterization of Inhomogeneous Lipschitz Spaces31

Exercises34

6.4 Hardy Spaces37

6.4.1 Definition of Hardy Spaces37

6.4.2 Quasinorm Equivalence of Several Maximal Functions40

6.4.3 Consequences of the Characterizations of Hardy Spaces53

6.4.4 Vector-Valued Hp and Its Characterizations56

6.4.5 Singular Integrals on Hardy Spaces58

6.4.6 The Littlewood-Paley Characterization of Hardy Spaces63

Exercises66

6.5 Besov-Lipschitz and Triebel-Lizorkin Spaces68

6.5.1 Introduction of Function Spaces68

6.5.2 Equivalence of Definitions71

Exercises76

6.6 Atomic Decomposition78

6.6.1 The Space of Sequences Fα,q p78

6.6.2 The Smooth Atomic Decomposition of Fα,q p78

6.6.3 The Nonsmooth Atomic Decomposition of Fα,q p82

6.6.4 Atomic Decomposition of Hardy Spaces86

Exercises90

6.7 Singular Integrals on Function Spaces93

6.7.1 Singular Integrals on the Hardy Space H193

6.7.2 Singular Integrals on Besov-Lipschitz Spaces96

6.7.3 Singular Integrals on Hp(Rn)96

6.7.4 A Singular Integral Characterization of H1(Rn)104

Exercises111

7 BMO and Carleson Measures117

7.1 Functions of Bounded Mean Oscillation117

7.1.1 Definition and Basic Properties of BMO118

7.1.2 The John-Nirenberg Theorem124

7.1.3 Consequences of Theorem 7.1.6128

Exercises129

7.2 Duality between H1 and BMO130

Exercises135

7.3 Nontangential Maximal Functions and Carleson Measures135

7.3.1 Definition and Basic Properties of Carleson Measures136

7.3.2 BMO Functions and Carleson Measures141

Exercises144

7.4 The Sharp Maximal Function146

7.4.1 Definition and Basic Properties of the Sharp Maximal Function146

7.4.2 A Good Lambda Estimate for the Sharp Function148

7.4.3 Interpolation Using BMO151

7.4.4 Estimates for Singular Integrals Involving the Sharp Function152

Exercises155

7.5 Commutators of Singular Integrals with BMO Functions157

7.5.1 An Orlicz-Type Maximal Function158

7.5.2 A Pointwise Estimate for the Commutator161

7.5.3 Lp Boundedness of the Commutator163

Exercises165

8 Singular Integrals of Nonconvolution Type169

8.1 General Background and the Role of BMO169

8.1.1 Standard Kernels170

8.1.2 Operators Associated with Standard Kernels175

8.1.3 Calderón-Zygmund Operators Acting on Bounded Functions179

Exercises181

8.2 Consequences of L2 Boundedness182

8.2.1 Weak Type(1,1)and Lp Boundedness ofSingular Integrals183

8.2.2 Boundedness of Maximal Singular Integrals185

8.2.3 H1→L1 and L∞→BMO Boundedness of Singular Integrals188

Exercises191

8.3 The T(1)Theorem193

8.3.1 Preliminaries and Statement of the Theorem193

8.3.2 The Proof of Theorem 8.3.3196

8.3.3 An Application209

Exercises211

8.4 Paraproducts212

8.4.1 Introduction to Paraproducts212

8.4.2 L2 Boundedness of Paraproducts214

8.4.3 Fundamental Properties of Paraproducts216

Exercises222

8.5 An Almost Orthogonality Lemma and Applications223

8.5.1 The Cotlar-Knapp-Stein Almost Orthogonality Lemma224

8.5.2 An Application227

8.5.3 Almost Orthogonality and the T(1)Theorem230

8.5.4 Pseudodifferential Operators233

Exercises236

8.6 The Cauchy Integral of Calderón and the T(b) Theorem238

8.6.1 Introduction of the Cauchy Integral Operator along a Lipschitz Curve239

8.6.2 Resolution of the Cauchy Integral and Reduction of Its L2 Boundednessto a Quadratic Estimate242

8.6.3 A Quadratic T(1)Type Theorem246

8.6.4 A T(b) Theorem and the L2 Boundedness of the Cauchy Integral250

Exercises253

8.7 Square Roots of Elliptic Operators256

8.7.1 Preliminaries and Statement of the Main Result256

8.7.2 Estimates for Elliptic Operators on Rn257

8.7.3 Reduction to a Quadratic Estimate260

8.7.4 Reduction to a Carleson Measure Estimate261

8.7.5 The T(b) Argument267

8.7.6 The Proof of Lemma 8.7.9270

Exercises275

9 Weighted Inequalities279

9.1 The Ap Condition279

9.1.1 Motivation for the Ap Condition280

9.1.2 Properties of Ap Weights283

Exercises291

9.2 Reverse H？lder Inequality and Consequences293

9.2.1 The Reverse H？lder Property of Ap Weights293

9.2.2 Consequences of the Reverse H？lder Property297

Exercises299

9.3 The A∞ Condition302

9.3.1 The Class of A∞ Weights302

9.3.2 Characterizations of A∞ Weights304

Exercises308

9.4 Weighted Norm Inequalities for Singular Integrals309

9.4.1 A Review of Singular Integrals309

9.4.2 A Good Lambda Estimate for Singular Integrals310

9.4.3 Consequences of the Good Lambda Estimate316

9.4.4 Necessity of the Ap Condition321

Exercises322

9.5 Further Properties of Ap Weights324

9.5.1 Factorization of Weights324

9.5.2 Extrapolation from Weighted Estimates on a Single Lp0325

9.5.3 Weighted Inequalities Versus Vector-Valued Inequalities332

Exercises335

10 Boundedness and Convergence of Fourier Integrals339

10.1 The Multiplier Problem for the Ball340

10.1.1 Sprouting of Triangles340

10.1.2 The counterexample343

Exercises350

10.2 Bochner-Riesz Means and the Carleson-Sj？lin Theorem351

10.2.1 The Bochner-Riesz Kernel and Simple Estimates351

10.2.2 The Carleson-Sj？lin Theorem354

10.2.3 The Kakeya Maximal Function359

10.2.4 Boundedness of a Square Function361

10.2.5 The Proof of Lemma 10.2.5363

Exercises366

10.3 Kakeya Maximal Operators368

10.3.1 Maximal Functions Associated with a Set of Directions368

10.3.2 The Boundedness of ？ΣN on Lp(R2)370

10.3.3 The Higher-Dimensional Kakeya Maximal Operator378

Exercises384

10.4 Fourier Transform Restriction and Bochner-Riesz Means387

10.4.1 Necessary Conditions for Rp→q(Sn-1) to Hold388

10.4.2 A Restriction Theorem for the Fourier Transform390

10.4.3 Applications to Bochner-Riesz Multipliers393

10.4.4 The Full Restriction Theoremon R2396

Exercises402

10.5 Almost Everywhere Convergence of Bochner-Riesz Means403

10.5.1 A Counterexample for the Maximal Bochner-Riesz Operator404

10.5.2 Almost Everywhere Summability of the Bochner-Riesz Means407

10.5.3 Estimates for Radial Multipliers411

Exercises419

11 Time-Frequency Analysis and the Carleson-Hunt Theorem423

11.1 Almost Everywhere Convergence of Fourier Integrals423

11.1.1 Preliminaries424

11.1.2 Discretization of the Carleson Operator428

11.1.3 Linearization of a Maximal Dyadic Sum432

11.1.4 Iterative Selection of Sets of Tiles with Large Mass and Energy434

11.1.5 Proof of the Mass Lemma 11.1.8439

11.1.6 Proof of Energy Lemma 11.1.9441

11.1.7 Proof of the Basic Estimate Lemma 11.1.10446

Exercises452

11.2 Distributional Estimates for the Carleson Operator456

11.2.1 The Main Theorem and Preliminary Reductions456

11.2.2 The Proof of Estimate(11.2.8)460

11.2.3 The Proof of Estimate(11.2.9)462

11.2.4 The Proof of Lemma 11.2.2463

Exercises474

11.3 The Maximal Carleson Operator and Weighted Estimates475

Exercises479

Glossary483

References487

Index501